Simple-minded systems, configurations and mutations for representation-finite self-injective algebras

TL;DR

This paper studies simple-minded systems in stable module categories of representation-finite self-injective algebras, showing they can be classified and constructed via stable equivalences, mutations, and derived category lifts.

Contribution

It proves all simple-minded systems are images of simple modules under stable equivalences and can be generated algorithmically through mutations.

Findings

All simple-minded systems are images of simple modules under stable equivalences.

Simple-minded systems can be lifted to Nakayama-stable collections in the derived category.

Algorithms exist to construct all simple-minded systems via mutations.

Abstract

Simple-minded systems of objects in a stable module category are defined by common properties with the set of simple modules, whose images under stable equivalences do form simple-minded systems. Over a representation-finite self-injective algebra, it is shown that all simple-minded systems are images of simple modules under stable equivalences of Morita type, and that all simple-minded systems can be lifted to Nakayama-stable simple-minded collections in the derived category. In particular, all simple-minded systems can be obtained algorithmically using mutations.